The present application relates generally to an improved data processing apparatus and method and more specifically to mechanisms for minimizing uncertainty envelopes in trajectories of evolving ensemble members.
In many problems such as parameter and state estimation for dynamical systems, time series prediction, and functions approximation, one must estimate some unknown variable using available data. The data are always associated with some uncertainty, and it is necessary to evaluate how this uncertainty affects the estimated variables.
Many industries have applications in which it is important to predict the behavior of an ensemble of objects over time. For example, one may be attempting to predict the failure time of a device, health impact on a patient, failure to meet design specifications, diffusion of pollutant particles etc. In such applications, one studies the state of the ensemble of objects over time by employing a model. One studies the model in an abstract space (i.e., the state space) that contains trajectories of the members of the ensemble.
Since the future values of parameters of the objects are uncertain, analysis of the states of the ensemble includes enclosing the trajectories in uncertainty envelopes. Predicting the maximum extent of the uncertainty envelopes over time is important in order to know critical values for the applications.
Algorithms have been developed to predict the maximum envelope in state space that encompasses such an ensemble under an assumed model for the starting state of the ensemble and in the presence of noise and uncertainty. The minimax approach is one of many classical ways to pose a state estimation problem. More details on the minimax framework, set-membership uncertainty estimation and reachability analysis can be found in the following references: M. Milanese and R. Tempo, “Optimal algorithms theory for robust estimation and prediction,” IEEE Trans. Automat. Control, vol. 30, no. 8, pp. 730-738, 1985; F. L. Chernousko, State Estimation for Dynamic Systems. Boca Raton, Fla.: CRC, 1994; A. Nakonechny. “A minimax estimate for functionals of the solutions of operator equations.” Arch. Math. (Brno), vol. 14, no. 1, pp. 55-59, 1978; and, A. Kurzhanski and I. Vàlyi, Ellipsoidal calculus for estimation and control, ser. Systems & Control: Foundations & Applications. Boston, Mass.: Birkhaüser Boston Inc., 1997.